Diagonal hyperbolic systems with large and monotone data Part I: Global continuous solutions

In this paper, we study diagonal hyperbolic systems in one space dimension. Based on a new gradient entropy estimate, we prove the global existence of a continuous solution, for large and non-decreasing initial data. We remark that these results cover the case of systems which are hyperbolic but not strictly hyperbolic. Physically, this kind of diagonal hyperbolic systems appears naturally in the modelling of the dynamics of dislocation densities.